Physics: Work, Power and Energy

In one line: JEE/NEET Physics · Work, Energy & Power series · Part 1 of 8 · All parts →✪ Key points — the 30-second versionWork = force × distance moved ALONG the.

In fact, JEE/NEET Physics · Work, Energy & Power series · Part 1 of 8 · All parts →

✪ Key points — the 30-second version

Moreover, work = force × distance moved ALONG the force's direction
Push at an angle? Meanwhile, only the part of the force along the motion counts: W = Fd·cosθ
Meanwhile, carry a bag while walking on flat ground: work by you on the bag = ZERO (surprise!)
As a result, work can be negative — when the force fights the motion (friction)
In other words, unit: joule (J) = one newton pushing through one metre

Notably, push a wall all day — exhausting, but physics says you did ZERO work. Meanwhile, push a trolley the same effort — physics says you did real work. In fact, in physics, 'work' isn't sweat — it's force achieving movement along its own direction . Part 1 of the Work, Energy & Power series — the card that powers the whole chapter.

In this card

Indeed, the simple idea: force × movement
What each letter means
Specifically, the angle rule: only the matching part counts
The zero-work surprises
Positive and negative work
Solved examples
Common mistakes
Similarly, this physics in your daily life
Practice set
Recap

The Simple Idea: Force × Movement

Overall, work measures transfer of energy by a force. Meanwhile, the formula is almost embarrassingly simple — how hard you push × how far the thing moves in the direction you push . Moreover, one newton of push through one metre = one joule of work. But the strictness hides two traps, and both surprise everyone.

What Each Letter Means

Work as energy flow: force + movement along it → energy transferred (positive in, negative out — friction's arrow points backwards)

YOUR PUSHF along motion

W = F d cosθpositive work

object's KEincreases

friction: force OPPOSES motion
→ negative work → KE decreases

W = F × d × cos(angle)force × distance × the cosine of the angle between push-direction and motion-direction

Letter	What it means (plain words)	Value / unit
W	Consequently, work done — energy transferred by the force	joules (J)
F	the force applied	newtons (N)
d	Furthermore, distance the object MOVES (not how hard you tried!)	metres
angle	Likewise, between the force's direction and the motion's direction	In short, 0° = full work; 90° = zero; 180° = negative

The Angle Rule: Only the Matching Part Counts

Subsequently, pull a trolley with a slanted rope: only the forward part of your pull moves the trolley forward. Meanwhile, the upward part just lightens it (no forward movement from that). Therefore, cos(angle) keeps exactly the matching part: at 0° (pulling straight along) cos = 1, full work. At 60°, half your force counts; at 90°, cos = 0, nothing counts.

The Zero-Work Surprises

In fact, surprise 1 — pushing a wall: huge force, zero movement → d = 0 → work = 0. Indeed, your muscles burn stored energy (that's biology), but no work is done ON the wall.

Moreover, surprise 2 — carrying a bag on flat ground: your upward hold-force is perpendicular to your forward walking → angle = 90° → cos = 0 → work by the holding force = zero. Meanwhile, the bag moves horizontally; your force points up; they don't match.

In other words, surprise 3 — an orbiting satellite: gravity pulls toward Earth; motion is along the orbit. Indeed, for a circular orbit they're exactly perpendicular — gravity does zero work on a circular orbit. That's why the ISS never slows down.

Positive and Negative Work

Notably, force along motion (0°): positive work — energy given. Force against motion (180°, like friction on a sliding box): negative work — energy taken away. The sign is bookkeeping of energy flow, and it decides entire questions.

Solved Examples

✎ Easy — the pull. A 50 N pull along the ground drags a box 4 m. Work?

Indeed, angle 0°, cos = 1: W = 50 × 4 = 200 J.

Specifically, feel it: 200 J ≈ the energy of a phone charger for a second — modest, sensible.

Answer: 200 J

✎ Exam level — the slanted pull. The same 50 N pull at 60° above the ground, box still moves 4 m horizontally. Work by the pull?

Similarly, only the forward part counts: forward force = 50 × cos60° = 25 N.

Overall, w = 25 × 4 = 100 J — exactly half. The other 25 N (upward part) did zero work.

Answer: 100 J

✎ JEE level — friction's negative bookkeeping. A 20 kg box slides 5 m across a floor (grip μ = 0.3, g = 10) and stops. Work by friction?

Friction fights the motion → 180° → negative work.

Size: friction = 0.3 × 200 = 60 N.

Meaning: the box handed 300 J to the floor as heat — the energy bookkeeping balances.

Answer: −300 J (energy removed)

⚠ Mistakes students make — and how to avoid them

Confusing effort with work. Pushing a wall: effort yes, work no — the wall doesn't move. Physics asks what moved, not how tired you are.
Forgetting the angle. The force's full size goes into the formula only if it points along the motion. Otherwise multiply by cos(angle).
Carrying a bag = work? On flat ground, no — hold-force is perpendicular to walking. (On stairs, yes: your force gains a lifting component.)
Dropping minus signs. Friction on a moving box does NEGATIVE work; writing it positive breaks every later energy equation.

This Physics in Your Daily Life

◎ This physics in your daily life

Your electricity bill is in kilowatt-HOURS — 3.6 million joules each: the power company bills you for work done by your appliances.
Gym reality check: holding a plank does (almost) zero physics work — the burn is biology's cost of muscle tension, not work on a load. Moving the weight is where physics work happens.
A satellite never needs fuel to keep orbiting because gravity's work on a circular orbit is zero — the ISS falls around Earth for free, forever.
Cycling on flat ground is cheap; hills are expensive — on the flat your push mostly fights air; on a climb you do real lifting work (mgh every metre up, Part 3).
Regenerative braking in EVs reverses friction's role: the motor does negative work on the wheels and hands the energy to the battery instead of the brakes.

Practice set (answers hidden — try first)

(NEET-level) A 100 N box is lifted 2 m straight up. Work by the lifting force:

W = 100 × 2 × cos0° = 200 J.

(NEET-level) A porter carries a 20 kg load 50 m on flat ground. Work by the holding force:

Force (up) ⊥ motion (horizontal) → zero.

(JEE Main-level) A 40 N force at 60° drags a body 10 m horizontally:

W = 40 × 10 × cos60° = 200 J.

(Concept) Work done by Earth's gravity on the ISS in one circular lap:

Zero — gravity is perpendicular to the orbital motion.

(NEET-level) Friction of 25 N acts on a box sliding 8 m. Work by friction:

Opposes motion → 25 × 8 with negative sign = −200 J.

🧠 Memory tricks & everyday anchors — the 20-second revision

🧠 Chant: 'no movement, no work — no match, no work'.
🧠 90° = zero: perpendicular forces never do work — carry a bag, hold a wall, circular orbit.
🏠 Daily: your electricity bill literally counts joules — 1 unit = 3.6 MJ of work by your appliances.
🏠 Daily: EV regenerative braking flips friction's negative work into battery charge.
🔁 W = Fd·cosθ; joules; positive/negative by direction
🔁 perpendicular force → zero work, always
🔁 friction on a moving body does negative work

One idea, three doors — open whichever clicks for you

Same concept (what 'work' actually means in physics), three different ways of seeing it. If one door confuses you, try the next — at least one will stick.

Door 1 · The story way

Carry a heavy box across a flat room: exhausting, but physics says zero work — the box moved sideways, your force pointed up. Work is only counted when force and movement point TOGETHER. Effort isn't work; displacement in the force's direction is.

Door 2 · The numbers way

Lift 10 N by 2 m: W = 10×2 = 20 J. Carry the same box 5 m horizontally: gravity-work = zero (force up, motion sideways). Push 20 N into a wall until you sweat: wall doesn't move, work = 0. The metre-tape, not the muscles, gets the final word.

Door 3 · The picture way

Draw an arrow for force and an arrow for displacement from the same point; work is the shaded parallelogram between them. Aligned arrows: full box of shading. Perpendicular: no shading, no work. Opposite: shading counted negative — work done AGAINST you.

Why is this happening at all? Why only the aligned part? Because force is a push in a specific direction, and only motion along that push can be 'driven' by it. The perpendicular component of motion was paid for by something else — a different force. Splitting motion along force-lines is just honest accounting of who caused what.

▶ Recap card — save for revision week

W = Fd·cos(angle) — force × distance × matching fraction
0°: full work · 90°: zero · 180°: negative
zero-work surprises: pushing a wall, carrying a bag on flat ground, circular orbits
negative work = energy taken away (friction on a sliding box)
joule = newton through one metre

← Previous series: Rotational Motion Part 2: Kinetic Energy and the Work-Energy Theorem: The Great Shortcut →

Frequently Asked Questions

What should you know about The Simple Idea: Force × Movement?

Work measures transfer of energy by a force. The formula is almost embarrassingly simple — how hard you push × how far the thing moves in the direction you push . One newton of push through one metre = one joule of work. But the strictness hides two traps, and both surprise everyone.

What should you know about The Angle Rule: Only the Matching Part Counts?

Pull a trolley with a slanted rope: only the forward part of your pull moves the trolley forward. The upward part just lightens it (no forward movement from that). cos(angle) keeps exactly the matching part: at 0° (pulling straight along) cos = 1, full work. At 60°, half your force counts; at 90°, cos = 0, nothing counts.

What should you know about The Zero-Work Surprises?

Surprise 1 — pushing a wall: huge force, zero movement → d = 0 → work = 0. Your muscles burn stored energy (that's biology), but no work is done ON the wall. Surprise 2 — carrying a bag on flat ground: your upward hold-force is perpendicular to your forward walking → angle = 90° → cos = 0 → work by the holding force = zero. The bag moves horizontally; your force points up; they don't match.

What should you know about Positive and Negative Work?

Force along motion (0°): positive work — energy given. Force against motion (180°, like friction on a sliding box): negative work — energy taken away. The sign is bookkeeping of energy flow, and it decides entire questions.

What should you know about Solved Examples?

Angle 0°, cos = 1: W = 50 × 4 = 200 J. Feel it: 200 J ≈ the energy of a phone charger for a second — modest, sensible. ✔ Confusing effort with work. Pushing a wall: effort yes, work no — the wall doesn't move. Physics asks what moved, not how tired you are.

Kinetic Energy and the Work-Energy Theorem

In one line: Kinetic Energy and the Work-Energy Theorem — exam-ready notes in one glance.

In one line: JEE/NEET Physics · Work, Energy & Power series · Part 2 of 8 · All parts →✪ Key points — the 30-second versionMoving energy: KE = ½mv² — double the speed,.

In fact, JEE/NEET Physics · Work, Energy & Power series · Part 2 of 8 · All parts →

✪ Key points — the 30-second version

Moving energy: KE = ½mv² — double the speed, QUADRUPLE the energy
Therefore, the theorem: total work on a body = change in its moving energy
Meanwhile, why it’s a shortcut: no forces over time needed — just before and after
As a result, stopping distance grows as speed SQUARED — the road-safety physics
In other words, same speed, different mass: KE scales with mass; same mass: KE scales with speed²

Notably, a car at 60 km/h needs four times the distance to stop as at 30 km/h. Not twice — four times. In fact, that single fact kills accidents. Meanwhile, it comes from one formula: moving energy grows with the SQUARE of speed. In fact, part 2 of the Work, Energy & Power series .

In this card

Moving energy, simply
What each letter means
Indeed, the work-energy theorem: the great shortcut
Specifically, the v² law and road safety
Solved examples
Common mistakes
Similarly, this physics in your daily life
Practice set
Recap

Moving Energy, Simply

Moreover, anything moving carries energy of motion — kinetic energy (KE). Meanwhile, it depends on mass and speed, but NOT equally: mass counts once, speed counts twice (squared) :

KE = ½ × mass × speed²½mv² — the square is the whole personality

Letter	What it means (plain words)	Value / unit
KE	Therefore, kinetic energy — energy of motion	joules (J)
m	mass of the moving thing	kg
v	its SPEED	m/s — always squared here

Meanwhile, feel the square: a 50 kg cyclist at 10 m/s has 2,500 J. Meanwhile, at 20 m/s (only double) — 10,000 J, four times . Moreover, triple the speed, nine times the energy.

The Work-Energy Theorem: The Great Shortcut

total work on a body = its KE change (W_total = ½mv² − ½mu²)add up all the work (positive and negative) — that’s exactly how much the moving energy changed

As a result, why this is gold: Newton’s method needs force at every instant. Indeed, the theorem doesn’t — it only needs before and after speeds, whatever complicated path connected them. Therefore, a curved water slide, a bumpy road, a rollercoaster: total work is the same. The theorem skips every detail in between.

The v² Law and Road Safety

In other words, stopping means removing all the KE. Meanwhile, brakes remove energy roughly at a steady rate over distance. KE ∝ v², so stopping distance ∝ speed² . Meanwhile, 30→60 km/h: energy ×4, distance ×4. This one line of physics explains speed limits, school-zone signs. Why ‘he was only a bit faster’ is never true.

Solved Examples

✎ Easy — the cyclist. A 50 kg cyclist at 10 m/s. KE? And at 20 m/s?

At 10: ½ × 50 × 100 = 2,500 J. At 20: ½ × 50 × 400 = 10,000 J.

Notably, double speed, ×4 energy — the square, felt.

Answer: 2,500 J → 10,000 J

✎ Exam level — the theorem in action. A 1,000 kg car at 20 m/s brakes to a stop. Total work by brakes?

KE change: 0 − ½(1000)(400) = −200,000 J.

Indeed, theorem: work by brakes = −200 kJ — the brakes REMOVED 200 kJ (as heat. Meanwhile, brake discs glow on F1 cars for exactly this reason).

Specifically, follow-up: at 40 m/s the same car needs −800 kJ — and four times the stopping distance.

Answer: W = −200 kJ (removed)

✎ JEE level — the slide. A child (30 kg) slides from rest down a frictionless 4 m slide. Speed at the bottom (g = 10)?

Similarly, theorem route (no forces needed): gravity’s work = mgh = 30×10×4 = 1,200 J (the normal push does zero work — perpendicular). KE goes 0 → 1,200 J.

Overall, ½(30)v² = 1,200 → v² = 80 → v ≈ 8.9 m/s.

Consequently, notice: the slide’s shape never entered — curved, straight, wavy: same answer. That’s the theorem’s power.

Answer: v ≈ 8.9 m/s, whatever the slide’s shape

⚠ Mistakes students make — and how to avoid them

Furthermore, using speed without squaring ‘just for this once’. Every KE question punishes it. Write ½mv² and square first.
Likewise, confusing momentum (mv) with energy (½mv²). Meanwhile, doubling speed doubles momentum but quadruples energy — different quantities, different questions.
In short, forgetting negative work in the theorem. Meanwhile, friction’s work enters with a minus; total means total.
Subsequently, believing heavier needs more stopping distance. Meanwhile, at the same speed, a truck has more KE AND more braking force (roughly proportional) — stopping distance is nearly mass-independent; SPEED decides it.

This Physics in Your Daily Life

◎ This physics in your daily life

In fact, speed limits are energy laws: hitting a wall at 60 vs 30 km/h delivers four times the energy to your body — the entire case for slower zones.
F1 brake discs glow red-hot: they absorb a race car’s KE as heat. Meanwhile, ~1 MJ per hard stop — a kettle’s worth of boiling energy, in a second.
Overall, cyclists and pedestrians survive at 30, not 60: crash-energy quadruples with doubling speed — the physics behind every city speed limit debate.
Consequently, a hammer drives nails: swing energy ½mv² concentrated into the nail’s tiny stopping distance — enormous force from modest energy.
Furthermore, water dams and kettle elements, wind turbines and your leg muscles: all trade work for KE and back — this card is the exchange rate.

Practice set (answers hidden — try first)

(NEET-level) A 2 kg ball at 3 m/s. KE:

½ × 2 × 9 = 9 J.

(JEE Main-level) Speed tripled. KE becomes:

3² = 9 → nine times.

(NEET-level) A 500 kg bike at 20 m/s brakes to rest. Work by brakes:

0 − ½(500)(400) = −100 kJ.

(Concept) A box slides down a frictionless curved slide of height h. Bottom speed depends on:

Only h — the shape is irrelevant (work-energy theorem).

(JEE Main-level) Equal KE, masses 1 kg and 4 kg. Speed ratio:

½(1)v₁² = ½(4)v₂² → v₁ = 2v₂ → 2 : 1.

🧠 Memory tricks & everyday anchors — the 20-second revision

Likewise, 🧠 Chant: ‘double the speed, four times the trouble’.
🧠 Theorem shortcut: ‘forget the path, compare the speeds’.
🧠 Perpendicular = free: slides, orbits, carrying bags — no work, no KE change from those forces.
🏠 Daily: every speed-limit sign is ½mv² wearing a red circle.
🏠 Daily: glowing F1 brakes — a car’s KE converted visibly to heat.
🔁 KE = ½mv², joules
🔁 W_total = ΔKE — the before/after shortcut
🔁 stopping distance ∝ v²

One idea, three doors — open whichever clicks for you

Same concept (why kinetic energy is the work-energy shortcut), three different ways of seeing it. If one door confuses you, try the next — at least one will stick.

Door 1 · The story way

To find how fast a sled goes at the bottom of a hill, you could track force at every instant — or just audit the books: all the work done = the bank balance of motion (½mv²). Work-energy theorem says the ugly path-by-path calculation always collapses to one subtraction.

Door 2 · The numbers way

Push a 2 kg sled with 16 N over 4 m (frictionless): W = 64 J. Then ½(2)v² = 64 → v = 8 m/s. No timing, no acceleration formulas — one line. Add friction taking 16 J: net 48 J, v = 6.9 m/s. The theorem absorbed everything.

Door 3 · The picture way

Picture energy as water: work pours it in, friction drains it out, and the level of the tank is ½mv². However complicated the pouring and draining, you only read the final level. Speed is just the tank’s height, converted.

Why is this happening at all? Why does ½ appear and why squared? Because a force accelerating a mass does so while the mass keeps moving — the faster it already goes, the more distance the force acts over per unit of extra speed. That compounding makes energy grow as speed-squared; the ½ is the leftover of integrating that growth honestly.

▶ Recap card — save for revision week

KE = ½mv² — speed squared: double speed, ×4 energy
work-energy theorem: total work = KE change — before/after only
normal force and other perpendicular forces do zero work
stopping distance ∝ speed² — the road-safety law
momentum ∝ v; energy ∝ v² — different books

← Part 1: Work Done: When a Force Actually Achieves Something Part 3: Potential Energy: Stored Work, Ready to Strike →

Frequently Asked Questions

What should you know about Moving Energy, Simply?

Anything moving carries energy of motion — kinetic energy (KE). It depends on mass and speed, but NOT equally: mass counts once. Speed counts twice (squared) : Feel the square: a 50 kg cyclist at 10 m/s has 2,500 J. At 20 m/s (only double) — 10,000 J, four times . Triple the speed, nine times the energy.

What should you know about The Work-Energy Theorem: The Great Shortcut?

Why this is gold: Newton’s method needs force at every instant. The theorem doesn’t — it only needs before and after speeds, whatever complicated path connected them. A curved water slide, a bumpy road, a rollercoaster: total work is the same. The theorem skips every detail in between.

What should you know about The v² Law and Road Safety?

Stopping means removing all the KE, and brakes remove energy roughly at a steady rate over distance. KE ∝ v², so stopping distance ∝ speed² . 30→60 km/h: energy ×4, distance ×4. This one line of physics explains speed limits, school-zone signs. Why ‘he was only a bit faster’ is never true.

What should you know about Solved Examples?

Double speed, ×4 energy — the square, felt. ✔ Using speed without squaring ‘just for this once’. Every KE question punishes it. Write ½mv² and square first. Confusing momentum (mv) with energy (½mv²). Doubling speed doubles momentum but quadruples energy — different quantities, different questions.

What should you know about This Physics in Your Daily Life?

Speed limits are energy laws: hitting a wall at 60 vs 30 km/h delivers four times the energy to your body — the entire case for slower zones. F1 brake discs glow red-hot: they absorb a race car’s KE as heat. ~1 MJ per hard stop — a kettle’s worth of boiling energy, in a second.

Potential Energy: Stored Work, Ready to Strike

In one line: JEE/NEET Physics · Work, Energy & Power series · Part 3 of 8 · All parts →✪ Key points — the 30-second versionStored energy from position: lifted = height.

JEE/NEET Physics · Work, Energy & Power series · Part 3 of 8 · All parts →

✪ Key points — the 30-second version

Stored energy from position: lifted = height energy (mgh), stretched = spring energy (½kx²)
Gravity is 'conservative': path doesn't matter, only height change
PE is a debt/credit between states — changes matter, not absolutes
PE → KE freely: the pendulum, the rollercoaster, the hydro dam
Spring energy grows with stretch SQUARED: double the pull, ×4 stored

Lift a brick to a rooftop — it quietly stores the work you did. Therefore, Let it go — the stored energy returns as motion. Dams, rollercoasters, archer's bows: all run on this stored energy. Meanwhile, Potential energy (PE) is work banked by changing a position — height or stretch. Part 3 of the Work, Energy & Power series.

In this card

Height energy: mgh
Spring energy: ½kx²
What each letter means
Conservative: path never matters
The PE ↔ KE trade
Solved examples
Common mistakes
This physics in your daily life
Practice set
Recap

Height Energy: mgh

PE_gravity = mass × g × height (mgh)the work you did lifting — stored, waiting

Letter	What it means (plain words)	Value / unit
m	mass being lifted	kg
g	gravity strength at the surface	≈ 10 m/s² (9.8 precise)
h	height ABOVE YOUR CHOSEN ZERO LEVEL	metres — you choose the zero!

Lifting a 1 kg brick up 1 m costs ~10 J — banked as PE, returnable on release. Note h is measured from a zero level you choose (the ground, a table, a roof) — only PE changes are physical, so pick the most convenient zero and stay consistent.

Spring Energy: ½kx²

PE_spring = ½ × stiffness × stretch²½kx² — stretch squared, like speed squared in KE

Letter	What it means (plain words)	Value / unit
k	stiffness — how many newtons per metre of stretch	N/m (a stiff spring has big k)
x	stretch or squeeze from natural length	metres — always squared

Stretch a spring double: four times the stored energy. Compress triple: nine times. Consequently, The square makes spring energy grow brutally — why a fully compressed loaded spring is genuinely dangerous.

Conservative: Path Never Matters

Gravity has a beautiful property: the work it does depends ONLY on height change — not the route. Furthermore, Lift a brick straight up 2 m or carry it up a spiral ramp 2 m: gravity's PE change is identical (mgh both ways). Forces like this are called conservative; they allow the free PE↔KE trading of Part 4. However, Friction is the opposite: its work depends on path length — energy leaks away and never returns.

The PE ↔ KE Trade

Drop the brick: PE (mgh) converts to KE (½mv²). Moreover, Swing a pendulum: height energy ↔ motion energy, back and forth. Stretch a bow: your work → spring PE → arrow KE. In fact, All day, the universe trades between stored and moving energy — Part 4 makes the ledger exact.

Solved Examples

✎ Easy — the brick. A 2 kg brick lifted 5 m (g = 10). PE stored?

Direct: mgh = 2 × 10 × 5 = 100 J.

Check: 100 J banked = the KE it will have falling back — 100 = ½(2)v² → v = 10 m/s. ✔

Answer: 100 J

✎ Exam level — the spring. A spring of stiffness 200 N/m stretched 10 cm. Stored energy? And at 20 cm?

At 10 cm: ½(200)(0.1²) = 1 J. At 20 cm: ½(200)(0.2²) = 4 J.

Double stretch = ×4 energy — the square again. ✔

Answer: 1 J → 4 J (double stretch, ×4)

✎ JEE level — combined store. A 0.5 kg ball is placed on a vertical spring (k = 500 N/m) compressed 20 cm, then released. Max height above the release point (g = 10)?

Energy ledger: spring PE → height PE. ½(500)(0.2²) = 10 J = mgh → 10 = 0.5 × 10 × h.

h = 2 m.

Check: spring gives back everything it stored (ideal spring) — 10 J lifts 0.5 kg by 2 m. ✔

Answer: rises 2 m

⚠ Mistakes students make — and how to avoid them

Forgetting to square the stretch in ½kx². The #1 spring error — always square x first (in metres!).
Mixing cm and m. 10 cm must be 0.10 m before any formula. Notably, Half the wrong answers in this chapter start as centimetres.
Changing the zero level mid-problem. Choose one zero for h and never move it.
Assuming friction trades like gravity. Only conservative forces (gravity, ideal springs) bank energy returnably — friction's losses are one-way.

This Physics in Your Daily Life

◎ This physics in your daily life

Hydroelectric dams are PE banks: rain lifts water (Sun's work), dams hold the height, turbines cash mgh back as electricity — most of the world's renewable power is literally stored height.
A ball-point pen's click, a car's suspension, a trampoline, your mattress springs: ½kx² giving you comfort or function every day.
Archery and slingshots: muscle work banked in a bent bow / stretched rubber, returned as arrow speed in a millisecond.
Rollercoasters lift you once (the clacking chain lift = charging mgh) then trade PE↔KE for the whole ride — no engine needed after the first hill.
Pumped-storage power stations buy cheap night electricity to pump water uphill, then sell it back as mgh at peak hours — a battery made of a lake.

Practice set (answers hidden — try first)

(NEET-level) A 5 kg bag on a 3 m table (zero at floor). PE:

5 × 10 × 3 = 150 J.

(JEE Main-level) Spring k = 800 N/m compressed 5 cm. Energy:

½ × 800 × (0.05)² = 1 J.

(Concept) Lifting a stone 2 m straight vs along a 5 m ramp (no friction): gravity's PE gain is:

Identical — 2mg in both cases (path-free).

(NEET-level) Doubling a spring's stretch multiplies stored energy by:

4 (x²).

(JEE Main-level) A 1 kg ball dropped from 20 m: KE at the ground (no air):

All mgh → 200 J → v = 20 m/s (zero level at ground).

🧠 Memory tricks & everyday anchors — the 20-second revision

🧠 Chant: 'height banks, springs bank — squares and products, then thank'.
🧠 Two squares rule: KE has v², spring PE has x² — the chapter's two squares.
🧠 Path-free: straight up or spiral up — gravity only counts the height.
🏠 Daily: every dam is a battery made of a lake — mgh as national infrastructure.
🏠 Daily: your mattress and car suspension cash ½kx² for you all night and every bump.
🔁 PE_gravity = mgh (choose one zero)
🔁 PE_spring = ½kx² — square first, metres only
🔁 conservative = path-independent (gravity, springs)

One idea, three doors — open whichever clicks for you

Same concept (what potential energy really stores), three different ways of seeing it. If one door confuses you, try the next — at least one will stick.

Door 1 · The story way

A stretched spring, a lifted brick, a held-breath dam: all are frozen accidents waiting to happen. Potential energy is work you already did, parked in shape or position, retrievable on demand. Nothing visible moves — the energy hides in geometry.

Door 2 · The numbers way

Lift 5 kg by 2 m: you paid mgh = 98 J. Let go: gravity repays all 98 J as kinetic energy (v = 6.3 m/s). Compress a spring 0.1 m with k = 200 N/m: ½kx² = 1 J parked. Same currency, different vaults.

Door 3 · The picture way

Draw a landscape of hills and valleys with a ball resting partway up a slope. Height above the valley floor = stored energy. Nudge the ball: it rolls down, trading height for speed, the total constant. The map IS the energy budget.

Why is this happening at all? Why is energy 'stored' at all? Because forces like gravity and springs are conservative: the work they do depends only on where you start and end, not the path. Path-independence means some function of position alone must exist to keep the books — and that function is potential energy. It isn't a substance; it's bookkeeping that's forced to exist.

▶ Recap card — save for revision week

height energy: mgh — work banked by lifting
spring energy: ½kx² — stretch SQUARED (metres!)
conservative forces: path irrelevant, only height/stretch change matters
choose one zero level and keep it
friction is non-conservative: energy leaks one-way

← Part 2: Kinetic Energy and the Work-Energy Theorem: The Great Shortcut Part 4: Conservation of Energy: The Universe's Perfect Bookkeeping →

Frequently Asked Questions

What should you know about Height Energy: mgh?

What should you know about Spring Energy: ½kx²?

Stretch a spring double: four times the stored energy. Compress triple: nine times. In other words, The square makes spring energy grow brutally — why a fully compressed loaded spring is genuinely dangerous.

What should you know about Conservative: Path Never Matters?

Gravity has a beautiful property: the work it does depends ONLY on height change — not the route. Specifically, Lift a brick straight up 2 m or carry it up a spiral ramp 2 m: gravity's PE change is identical (mgh both ways). Forces like this are called conservative; they allow the free PE↔KE trading of Part 4. Indeed, Friction is the opposite: its work depends on path length — energy leaks away and never returns.

What should you know about The PE ↔ KE Trade?

Drop the brick: PE (mgh) converts to KE (½mv²). In short, Swing a pendulum: height energy ↔ motion energy, back and forth. Stretch a bow: your work → spring PE → arrow KE. Similarly, All day, the universe trades between stored and moving energy — Part 4 makes the ledger exact.

What should you know about Solved Examples?

Direct: mgh = 2 × 10 × 5 = 100 J. Check: 100 J banked = the KE it will have falling back — 100 = ½(2)v² → v = 10 m/s. ✔ Forgetting to square the stretch in ½kx². Therefore, The #1 spring error — always square x first (in metres!). Mixing cm and m. 10 cm must be 0.10 m before any formula. Meanwhile, Half the wrong answers in this chapter start as centimetres.

Conservation of Energy: The Universe's Perfect Bookkeeping

In one line: JEE/NEET Physics · Work, Energy & Power series · Part 4 of 8 · All parts →✪ Key points — the 30-second versionTotal energy (KE + PE) stays constant when.

In fact, JEE/NEET Physics · Work, Energy & Power series · Part 4 of 8 · All parts →

✪ Key points — the 30-second version

Moreover, total energy (KE + PE) stays constant when only conservative forces act
Falling: mgh converts exactly to ½mv² — v = √(2gh), mass cancels!
Meanwhile, with friction: ME_lost = friction force × distance (the leak is measurable)
As a result, pendulum and rollercoaster: endless PE↔KE trading
In other words, energy is never destroyed — only moved or downgraded

Notably, drop anything — a feather (in vacuum) or an elephant — from the same height. Meanwhile, both hit the ground at the same speed. In fact, mass doesn’t even enter the answer. That’s energy conservation at work: the universe’s most reliable bookkeeping. Part 4 of the Work, Energy & Power series .

In this card

The one rule
What each letter means
Indeed, the famous result: v = √(2gh), no mass anywhere
When friction leaks the ledger
The pendulum’s endless trade
Solved examples
Common mistakes
Specifically, this physics in your daily life
Practice set
Recap

The One Rule

The energy see-saw: what motion loses, height gains — the total never changes (until friction leaks it as heat)

total = KE + PE (constant)
KEmax
PE 0
bottom of swing — all motion
KE 0
PEmax
top of swing — all stored

KE + PE = constant (when only gravity/springs act)motion energy + stored energy = unchanging total

Letter	What it means (plain words)	Value / unit
KE	motion energy ½mv²	J
PE	Similarly, stored energy: mgh (height) and/or ½kx² (spring)	J
friction (if present)	Overall, the leak: total drops by friction × distance	the only common spoiler

Consequently, read it as a see-saw: what KE loses, PE gains, exactly. Meanwhile, total never changes (with only gravity/springs). Moreover, with friction, the total still doesn’t vanish — it leaks out as heat : mechanical energy lost = friction force × distance slid.

The Famous Result: v = √(2gh), No Mass Anywhere

Furthermore, drop from height h: mgh = ½mv² → divide both sides by m — mass cancels completely → v = √(2gh). Meanwhile, heavy or light, same landing speed (in vacuum). Therefore, from 20 m: v = √400 = 20 m/s. From 45 m (with g = 10): 30 m/s. One line, no mass, no time — the most useful result in the chapter.

When Friction Leaks the Ledger

Likewise, real slides and roads have friction. Indeed, the bookkeeping then reads: (KE + PE)_start = (KE + PE)_end + friction × distance. Meanwhile, the leak isn’t lost — it’s heat (why brake discs glow, why rubbing warms hands). Questions love this: ‘how far does it slide before stopping?’ — the leak formula answers in one line.

The Pendulum’s Endless Trade

In short, a pendulum swings because energy endlessly converts: maximum height (all PE. Meanwhile, still) → bottom (all KE, fastest) → the other side’s height (all PE again). With zero friction it would swing forever. Real pendulums leak tiny heat each swing — that’s why clocks needed winding.

Solved Examples

✎ Easy — the drop. Speed after falling 45 m (no air, g = 10)?

In other words, famous result: v = √(2gh) = √(2 × 10 × 45) = √900.

Notably, v = 30 m/s — no mass needed, ever.

Answer: 30 m/s

✎ Exam level — the ramp with friction. A 2 kg block slides from rest down a 3 m ramp (angle: height = 1.5 m) with friction 4 N acting along a 3 m path. Speed at the bottom (g = 10)?

Indeed, ledger: start PE = 2 × 10 × 1.5 = 30 J. Leak = friction × distance = 4 × 3 = 12 J. Remaining for KE = 18 J.

Specifically, ½(2)v² = 18 → v = √18 ≈ 4.24 m/s.

Check: without friction it’d be √30 ≈ 5.48 — friction slowed it, as it must.

Answer: v ≈ 4.24 m/s

✎ JEE level — the loop. A bead slides from rest at height h on a frictionless track with a vertical loop of radius R at the bottom. Minimum h to complete the loop?

Two conditions meet: at the loop’s top, gravity supplies the needed centripetal push: mg = mv²/R → v²_top = gR. Energy: mg·h = mg·(2R) + ½m·gR → h = 2R + R/2.

h = 2.5R.

This is the classic rollercoaster design number — five-halves the loop radius (in practice more, for friction).

Answer: h = 2.5R (the rollercoaster rule)

⚠ Mistakes students make — and how to avoid them

Putting mass in the drop formula. v = √(2gh) has no mass — inserting one means the algebra was never finished.
Forgetting the friction leak term. ‘Energy is conserved’ is FALSE with friction present; mechanical energy falls by friction × distance.
Height measured inconsistently. Keep one zero level for the whole problem (Part 3’s rule).
Believing energy conservation means nothing is lost ever. Energy is never destroyed — but it DOWNGRADES to heat, which is usually unusable. The ledger always balances; usefulness doesn’t.

This Physics in Your Daily Life

◎ This physics in your daily life

Every rollercoaster’s first hill is its battery — the rest of the ride spends that mgh. Engineers add margin above 2.5R for friction.
Hydro dams again, quantitatively: 1,000 tonnes falling 100 m delivers ~1 billion joules — v = √(2gh) for the water, then turbines take over.
Regenerative braking: EVs intercept the KE you’d normally burn in brakes and bank it into the battery — conservation, monetised.
A swing in the park: you pump by leaning at the right moments (adding small energy each cycle). Friction and air take tiny tolls — the trade is visible physics.
Meteors burn up because v is enormous: ½mv² at 30 km/s converts to heat on air contact — conservation you can watch as a shooting star.

Practice set (answers hidden — try first)

(NEET-level) Speed after a 20 m free fall (g = 10):

√(2×10×20) = 20 m/s.

(JEE Main-level) A 1 kg block slides 5 m on flat ground against friction 6 N, starting at 8 m/s. It stops after:

KE = ½(1)(64) = 32 J = 6 × d → d ≈ 5.33 m.

(Concept) A pendulum’s speed is maximum at:

The lowest point — all PE traded into KE.

(JEE Main-level) Loop-the-loop minimum release height (loop radius R, frictionless):

2.5R.

(Concept) With friction present, ‘energy is conserved’ — true or false?

Mechanical energy: false (leaks as heat). Total energy including heat: always true.

🧠 Memory tricks & everyday anchors — the 20-second revision

🧠 Chant: ‘what motion loses, height gains — the total never changes’.
🧠 The mass-free line: ‘drop questions don’t need mass — √2gh and done’.
🧠 2.5R: the rollercoaster rule — say it like a phone number.
🏠 Daily: every coaster’s first hill is the ride’s battery; everything after is spending.
🏠 Daily: a shooting star is ½mv² turned to heat in front of your eyes.
🔁 KE + PE constant under gravity/springs
🔁 v = √(2gh): no mass, no time needed
🔁 friction leak = friction × distance → heat

One idea, three doors — open whichever clicks for you

Same concept (why energy conservation cannot fail), three different ways of seeing it. If one door confuses you, try the next — at least one will stick.

Door 1 · The story way

A ball bounces lower each time and stops — energy ‘lost’? No: it became heat in the floor, sound in the air, wobble in the ball. Energy never disappears; it only changes costume. Conservation isn’t a rule physics follows — it’s a rule geometry forces.

Door 2 · The numbers way

Drop 1 kg from 10 m: PE = 98 J. Just before impact: KE = 98 J. After a messy inelastic bounce: 30 J motion + 60 J heat + 8 J sound. The costumes changed; the total never left 98 J.

Door 3 · The picture way

Picture a set of water tanks connected by pipes labelled ‘kinetic’, ‘potential’, ‘heat’, ‘sound’, ‘light’. Water flows between tanks in every direction, but the pipes never leak outside the system. Read any moment: total level constant.

Why is this happening at all? Why is it IMPOSSIBLE to violate? Because energy conservation comes from time-symmetry (Noether’s theorem): physics today is the same as physics tomorrow, and that symmetry mathematically demands a conserved quantity. As long as the laws don’t change with time, energy cannot be created or destroyed — conservation is geometry in disguise.

▶ Recap card — save for revision week

KE + PE = constant (gravity/springs only)
v = √(2gh) — the mass-free drop formula
with friction: ME drops by friction × distance (it becomes heat)
pendulum/rollercoaster: endless PE↔KE trade
loop-the-loop minimum: start at 2.5R

← Part 3: Potential Energy: Stored Work, Ready to Strike Part 5: Power and Efficiency: How FAST You Can Do the Work →

Frequently Asked Questions

What should you know about The One Rule?

Read it as a see-saw: what KE loses, PE gains, exactly. Total never changes (with only gravity/springs). With friction, the total still doesn’t vanish — it leaks out as heat : mechanical energy lost = friction force × distance slid.

What should you know about The Famous Result: v = √(2gh), No Mass Anywhere?

Drop from height h: mgh = ½mv² → divide both sides by m — mass cancels completely → v = √(2gh). Heavy or light, same landing speed (in vacuum). From 20 m: v = √400 = 20 m/s. From 45 m (with g = 10): 30 m/s. One line, no mass, no time — the most useful result in the chapter.

What should you know about When Friction Leaks the Ledger?

Real slides and roads have friction. The bookkeeping then reads: (KE + PE)_start = (KE + PE)_end + friction × distance. The leak isn’t lost — it’s heat (why brake discs glow, why rubbing warms hands). Questions love this: ‘how far does it slide before stopping?’ — the leak formula answers in one line.

What should you know about The Pendulum's Endless Trade?

A pendulum swings because energy endlessly converts: maximum height (all PE, still) → bottom (all KE, fastest) → the other side’s height (all PE again). With zero friction it would swing forever. Real pendulums leak tiny heat each swing — that’s why clocks needed winding.

What should you know about Solved Examples?

Famous result: v = √(2gh) = √(2 × 10 × 45) = √900. v = 30 m/s — no mass needed, ever. ✔ Putting mass in the drop formula. v = √(2gh) has no mass — inserting one means the algebra was never finished. Forgetting the friction leak term. ‘Energy is conserved’ is FALSE with friction present; mechanical energy falls by friction × distance.

Power and Efficiency: How FAST You Can Do the Work

In one line: JEE/NEET Physics · Work, Energy & Power series · Part 5 of 8 · All parts →

In fact, JEE/NEET Physics · Work, Energy & Power series · Part 5 of 8 · All parts →

✪ Key points — the 30-second version

Power = work done per second (P = W/t) — the speed of energy transfer
Therefore, one watt = one joule per second; your household runs on kilowatts
Meanwhile, the two workhorse forms: P = Fv (force × speed) and P = mgh/t (lifting)
As a result, efficiency = useful output ÷ total input — nothing real is 100%
In other words, same energy in less time = more power

Notably, two students carry the same 20 kg load up the same stairs — identical work. Meanwhile, one takes 1 minute, the other takes 10 seconds. Same work, wildly different POWER. In fact, power is the speed of doing work — and it’s what engines, motors and athletes are actually rated in. Part 5 of the Work, Energy & Power series.

In this card

Power, simply
What each letter means
The two workhorse formulas
Efficiency: nothing is 100%
Solved examples
Common mistakes
Indeed, this physics in your daily life
Practice set
Recap

Power, Simply

Power = work ÷ time (P = W/t)how many joules per second — the pace of energy transfer

Work tells you how much energy was transferred. Power tells you how quickly it happened. Divide the work done by the time taken, and you get the rate of energy transfer — measured in watts.

Letter	What it means (plain words)	Value / unit
P	power	watts (W) = joules/second
W	work (or energy) delivered	joules
t	time taken	seconds

Specifically, feel the numbers: a phone charger draws ~20 W, a ceiling fan ~75 W, a microwave ~1,000 W. Meanwhile, a car at highway pace consumes ~20,000 W, and a cricket ball’s throw delivered in 0.1 s releases ~1,000 W momentarily. Moreover, horsepower (the unit on car specs) = 746 W.

A quick sanity check on units: 1 watt = 1 joule per second, so 1 kilowatt (kW) = 1,000 J/s. Household appliances are rated in kilowatts because a watt is simply too small a unit to be convenient at that scale. And when your electricity bill says “units,” it means kilowatt-hours — energy, not power — a distinction examiners love to test.

The Two Workhorse Formulas

P = force × speed (P = Fv) · lifting: P = mgh/tP = Fv: engines and motors; mgh/t: pumps, stairs, cranes

Where does P = Fv come from? Start with P = W/t. Since work W = F × d, we get P = Fd/t. But d/t is just speed v — so P = Fv. No new physics, just the definition rearranged into its most useful form.

Similarly, P = Fv is why cars struggle uphill: at fixed engine power, the greater force needed for climbing forces a lower speed. Meanwhile, it’s also why you slow down when cycling into a headwind — same legs (same power), more force needed, so speed must drop. The product F × v stays constant; force and speed trade off against each other.

The lifting form, P = mgh/t, is just P = W/t with W = mgh substituted in. Use it for any problem involving raising a mass: stair climbs, water pumps, cranes, elevators. In exam problems, identify which situation you have — steady force against resistance (use P = Fv) or lifting against gravity (use P = mgh/t) — and the formula choice makes itself.

Efficiency: Nothing Is 100%

efficiency = useful energy out ÷ total energy in (× 100%)always below 100% — the missing part becomes heat, sound or friction losses

No real machine delivers all the energy you feed it. Friction turns some into heat; vibration and sound carry some away; motors warm up as they run. Efficiency measures what fraction of the input actually does the job you wanted.

Overall, a car engine is ~25–35% efficient (most fuel energy leaves as heat through the radiator and exhaust). Indeed, an LED bulb converts ~40–50% of its energy to light (versus an old filament bulb’s ~5%); an electric motor manages ~85–95%. Therefore, efficiency questions are pure percentage bookkeeping — just keep the word “useful” crystal clear in your mind.

One habit that prevents most errors: write down explicitly which quantity is the useful output and which is the total input before you divide. Efficiency is always output ÷ input, and the answer must come out below 1 (or 100%).

Solved Examples

✎ Easy — the stair climb. A 60 kg student climbs 4 m of stairs in 10 s (g = 10). Average power?

Consequently, work: mgh = 60 × 10 × 4 = 2,400 J. Meanwhile, power: 2,400/10 = 240 W — about three ceiling fans’ worth, sustained by the legs.

Answer: 240 W

✎ Exam level — the engine. A car engine delivers 40 kW at a steady 20 m/s. The driving force?

In other words, use P = Fv: F = 40,000/20 = 2,000 N.

Notably, check the physics: steady speed means this force exactly balances air + road resistance. Need more force (uphill)? Speed must fall — power is fixed.

Answer: 2,000 N

✎ JEE level — efficiency chain. A pump motor (efficiency 80%) fills a tank with 10,000 kg of water lifted 20 m in 500 s. Electric power drawn (g = 10)?

Indeed, useful output: mgh = 10,000 × 10 × 20 = 2 × 10⁶ J; useful power = 2 × 10⁶/500 = 4 kW.

Efficiency 80%: input = 4/0.8 = 5 kW drawn from the grid.

The missing 1 kW = motor heat — that’s exactly what the 80% figure meant.

Answer: 5 kW drawn from the grid

⚠ Mistakes students make — and how to avoid them

Confusing energy and power. A 100 W bulb used for 10 hours consumes 1,000 Wh = 1 unit of electricity — power × TIME is energy; bills charge for energy, not power.
Forgetting P = Fv needs steady speed (or use it as average force × average speed — be consistent).
Impossible efficiency — if your answer says 120%, you divided the wrong way. Output ÷ input, never the reverse.
Skipping unit conversion: horsepower × 746 = watts; hours × 3,600 = seconds — convert before dividing.

This Physics in Your Daily Life

◎ This physics in your daily life

Your electricity meter counts kilowatt-HOURS: power (kW) × time (hours) — the unit on every bill is literally this card in action.
Fan/AC star ratings are efficiency labels: same cooling, fewer watts — the same physics saving you money.
Cars advertise horsepower (1 hp = 746 W) — the rate at which the engine can deliver energy. “Torque × rpm” from our Rotational series is the same number in disguise.
Cycling into a headwind: your legs have fixed power. More drag force → less speed — P = Fv as lived experience.
Cricket fast bowlers: ~150 J delivered into a ball over ~0.1 s ≈ 1,500 W — sprinter-level power from a short run-up.
Climbing the same staircase twice as fast demands twice the average power — your body pays for speed with a racing heartbeat.

Practice Set (answers hidden — try first)

(NEET-level) A 50 kg person climbs 5 m in 25 s. Power (g = 10):

mgh/t = 2,500/25 = 100 W.

(JEE Main-level) An engine of 25 kW pushes a car at 10 m/s. Driving force:

F = P/v = 25,000/10 = 2,500 N.

(NEET-level) A 2 kW heater runs 3 hours. Energy consumed:

2 × 3 = 6 kWh = 6 units.

(JEE Main-level) A motor draws 5 kW to deliver 4 kW useful. Efficiency:

4/5 = 80%.

(Concept) At fixed engine power, climbing a hill, the car’s speed:

Falls — more force needed, and P = Fv stays fixed.

🧠 Memory tricks & everyday anchors — the 20-second revision

🧠 Chant: “work is how much, power is how fast.”
🧠 P = Fv: “fixed engine, hills eat speed.”
🧠 Bill math: “watts × hours = the unit on the meter.”
🏠 Daily: every appliance sticker in your kitchen is this card in print.
🏠 Daily: headwind cycling is P = Fv you can feel in your thighs.
🔁 P = W/t (watts); P = Fv; P = mgh/t for lifting
🔁 efficiency < 100%, always
🔁 1 hp = 746 W; 1 kWh = 3.6 MJ

One idea, three doors — open whichever clicks for you

Same concept (why power means how FAST work gets done), three different ways of seeing it. If one door confuses you, try the next — at least one will stick.

Door 1 · The story way

Two engines lift the same crane load to the same height: identical work done. One takes 10 s, one takes 1 s. Physics calls them different — power is the speed of doing work. Same salary, different pay rate.

Door 2 · The numbers way

Lift 100 kg by 10 m: W = mgh = 100 × 9.8 × 10 = 9,800 J. A 1-horsepower motor (746 W) does it in ~13 s. A 10 kW motor: 1 s. Your electricity bill counts kWh — power × time — literally billing you for total work delivered.

Door 3 · The picture way

Picture a graph of work done versus time: the height of the line is energy delivered; the SLOPE of the line is power. A steeper hill on the graph = a more powerful machine. A flat line = an idle one, no matter how strong.

Why is this happening at all? Why divide by time at all? Because work alone doesn’t describe capability: engines, muscles, and power plants are all limited not by total energy but by RATE — how fast they can convert fuel to motion. The rate is what sizes the machine, so the rate needs its own name.

▶ Recap card — save for revision week

P = W/t, watts = joules/second
P = Fv — the engine formula; more force = less speed at fixed power
lifting: P = mgh/t
efficiency = useful out ÷ total in — always < 100%
1 hp = 746 W; 1 kWh = 3.6 MJ (your bill’s unit)

← Part 4: Conservation of Energy: The Universe’s Perfect Bookkeeping Part 6: Collisions: The Great Sorting — What Survives, What Dies →

Frequently Asked Questions

What is power in physics, in simple words?

Power is the rate of doing work — how many joules of energy are transferred per second (P = W/t). Two machines can do identical work, but the one that finishes sooner is the more powerful one. That’s why power, not work, is what engines, motors and athletes are rated in.

What are the typical power ratings of everyday devices?

Feel the numbers: a phone charger draws ~20 W, a ceiling fan ~75 W, a microwave ~1,000 W. A car at highway pace consumes ~20,000 W, and a cricket ball’s throw delivered in 0.1 s releases ~1,000 W momentarily. Horsepower (the unit on car specs) = 746 W.

Why do cars slow down when climbing a hill?

P = Fv is the reason: at fixed engine power, the greater force needed for climbing forces a lower speed. It’s the same reason you slow down when cycling into a headwind — same legs (same power), more force needed, so speed must drop.

How efficient are common machines?

A car engine is ~25–35% efficient (most fuel energy becomes heat). An LED bulb converts ~40–50% to light (versus an old filament bulb’s ~5%); an electric motor manages ~85–95%. Efficiency questions are pure percentage bookkeeping — just keep the word “useful” clear: efficiency = useful output ÷ total input, always below 100%.

What is the most common power-and-efficiency exam mistake?

Confusing energy with power. Power is a rate (watts); energy is power multiplied by time (watt-hours or joules). A 100 W bulb used for 10 hours consumes 1,000 Wh = 1 unit of electricity — your bill charges for energy, not power. The second most common error: dividing input by output instead of output by input when calculating efficiency.

How does the electricity bill relate to power and energy?

Your electricity meter counts kilowatt-hours: power (kW) × time (hours). The “unit” on every bill is literally the physics of this card. Fan and AC star ratings are efficiency labels too — same cooling delivered, fewer watts consumed — the same physics saving you money every month.

Collisions: The Great Sorting — What Survives, What Dies

In one line: Collisions — what survives, what dies, and how to sort every crash into three types — exam-ready notes in one glance.

In one line: JEE/NEET Physics · Work, Energy & Power series · Part 6 of 8 · All parts →

JEE/NEET Physics · Work, Energy & Power series · Part 6 of 8 · All parts →

✪ Key points — the 30-second version

Momentum (mass × velocity) survives EVERY collision; kinetic energy often doesn’t
Elastic: both momentum and KE survive (ideal, bouncy)
Inelastic: momentum survives, KE partially dies (most real crashes)
Perfectly inelastic: bodies stick together — maximum KE loss
Explosions run the same rule in reverse

In every crash — a car wreck, a cricket ball striking a bat, two ice pucks colliding — one quantity ALWAYS survives the impact untouched, while another usually dies. Knowing which is which solves every collision question in one line each. This is Part 6 of the Work, Energy & Power series.

In this card

The survivor: momentum.
The casualty: kinetic energy.
The three types of collision.
Equal masses in elastic hits: the neat swap.
Solved examples.
Common mistakes.
This physics in your daily life.
Practice set.
Recap.

The Survivor: Momentum.

The collision sorting: momentum (blue) ALWAYS survives; kinetic energy (orange) dies in every real crash

2 kg
4 kg

6 m/s
BEFORE: momentum 12, KE 36 J

stuck together → 2 m/s
AFTER: momentum 12 ✓ KE 12 J (24 J → heat)

total (mass × velocity) before = total aftermomentum — the ‘quantity of motion’ — survives every collision, no exceptions

Why is momentum so unbreakable? It comes straight from Newton’s third-law bookkeeping. During the crash, the two bodies push on each other with equal and opposite forces for exactly the same time. Each body’s momentum changes by an equal and opposite amount, so the changes cancel perfectly. Whatever happens inside the crash — denting, heating, screeching, shattering — the total ‘quantity of motion’ stays locked. No exceptions, no conditions, no idealisations.

The Casualty: Kinetic Energy.

Unlike momentum, KE travels a one-way street: the crash can convert it into heat, dents, and sound — but never back. Once motion-energy becomes thermal jitter in bent metal, it is gone from the mechanical balance sheet for good. So the sorting of collisions is simple:

Type	Momentum	Kinetic energy	Example
Elastic	survives	survives TOO (ideal)	steel balls, gas molecules, billiards (near)
Inelastic	survives	partially dies	most real hits — cricket ball, cars
Perfectly inelastic	survives	maximum loss — bodies STICK	coupled train wagons, ball in mud

A note on language: ‘elastic’ is an idealisation. At everyday scales, gas molecules collide almost elastically, billiard balls collide very nearly elastically, but truly perfect elastic collisions exist only in textbooks. The exam trick is recognising which label a question intends — the word ‘stick’ or ‘combine’ always signals perfectly inelastic.

Equal Masses in Elastic Hits: The Neat Swap.

Here is a beautiful shortcut: equal masses colliding elastically simply exchange velocities. The moving one stops; the stopped one moves off with the first one’s speed. Newton’s cradle shows it in slow, clicking elegance — one ball in, one ball out, every time.

Why does the swap work? Solve the two conservation equations (momentum and KE) for equal masses and you’ll find the solutions m₁u₁ = m₂v₁ + m₁v₂ collapse to v₁ = 0, v₂ = u₁. No algebra needed in the exam — just remember: equal masses + elastic = velocity swap.

Solved Examples.

✎ Easy — the stick-together. A 2 kg ball at 6 m/s hits a stationary 4 kg ball; they stick. Common velocity?

Momentum survivor: (2×6) + 0 = (2+4)v → 12 = 6v.

v = 2 m/s.

Answer: 2 m/s

✎ Exam level — the energy audit. Same collision: how much KE died?

Before: ½(2)(36) = 36 J. After: ½(6)(4) = 12 J.

24 J died → heat and dent (67% of the energy!).

The pattern: momentum bookkeeping gave the speed; energy bookkeeping gives the loss. Always treat them as two separate questions.

Answer: 24 J lost (to heat/deformation)

✎ JEE level — elastic, unequal masses. A 1 kg ball at 4 m/s hits a stationary 3 kg ball elastically. Both final speeds?

Momentum: 1(4) = v₁ + 3v₂.

Elastic adds KE: 8 = ½v₁² + (3/2)v₂².

Solve the pair: v₁ = 4 − 3v₂ → substitute → v₂ = 2 m/s, v₁ = −2 m/s.

Read it: the light ball BOUNCES BACK (−2), the heavy one crawls forward (+2). Light things bounce off heavy things — a cricket ball off a bat’s massive swing, you off a truck.

Answer: 1 kg ball: −2 m/s (rebounds); 3 kg ball: +2 m/s

⚠ Mistakes students make — and how to avoid them

Conserving KE in a sticking collision. The classic error. Sticking = maximum energy death. Only momentum survives.
Ignoring sign conventions. All velocities must carry direction (+/−). ‘Both move at v’ when one actually reverses flips every answer.
Adding speeds instead of momenta. Mass × velocity, each with its sign — never velocities alone.
Forgetting the equal-mass elastic swap. It’s a free shortcut: equal masses exchange velocities — no algebra needed.

This Physics in Your Daily Life.

◎ This physics in your daily life

Car crumple zones are deliberate KE-killers: bending metal absorbs your crash energy over distance so your body doesn’t — momentum physics you hope never to use.
Newton’s cradle desk toy: the equal-mass elastic swap, clicked in metal, forever.
Cricket and tennis: ‘sweet spot’ hits are the most elastic (least energy lost to vibration) — the ball leaves fastest; off-center hits waste energy in sting.
Railway coupling is the perfectly-inelastic lab: wagons locking together at hump yards, sharing one speed after contact.
Airbags and helmets extend collision TIME — same momentum change, gentler force (the impulse idea) — safety engineering living inside this card.

Practice set (answers hidden — try first).

(NEET-level) 3 kg at 4 m/s sticks to a stationary 3 kg. Common speed:

(3×4)/6 = 2 m/s.

(JEE Main-level) That collision’s KE loss:

Before ½(3)(16) = 24 J; after ½(6)(4) = 12 J → 12 J lost.

(Concept) Two identical steel balls, one moving, one at rest, elastic collision. After:

Velocity swap — the first stops, the second moves at the original speed.

(JEE Main-level) A light ball hits a heavy wall elastically head-on. The ball’s speed:

Same speed, reversed direction (infinite-mass limit).

(Concept) Which quantity always survives a collision — and which can die even in an explosion-then-reassembly?

KE can die; momentum always survives.

🧠 Memory tricks & everyday anchors — the 20-second revision

🧠 Chant: ‘momentum always lives, energy often dies’.
🧠 Sticking = maximum death of KE — compute speed by momentum, loss by energy.
🧠 The swap: equal masses, elastic — they just trade velocities.
🏠 Daily: crumple zones are engineered energy death — protecting you by dying.
🏠 Daily: Newton’s cradle on a desk = this card as office jewellery.
🔁 momentum conservation: always
🔁 elastic: momentum + KE conserved
🔁 perfectly inelastic: stick together, max KE loss

One idea, three doors — open whichever clicks for you

Same concept (why collisions sort energy into ‘kept’ and ‘gone’), three different ways of seeing it. If one door confuses you, try the next — at least one will stick.

Door 1 · The story way

Billiards, car crashes, and mud hitting a wall are the same event with different honesty about energy. Elastic: the full kinetic balance survives. Inelastic: some motion-energy is spent permanently on deformation, heat, sound — never refunded.

Door 2 · The numbers way

Two 1 kg carts at 4 m/s meet head-on, elastic: both rebound at 4 m/s — total KE before 16 J, after 16 J. Same crash perfectly inelastic (they stick): both stop — 0 J of motion left, all 16 J gone as heat and dents. Momentum conserved in both; kinetic energy only in the first.

Door 3 · The picture way

Draw a bar chart split into ‘KE kept’ and ‘KE spent’ for three collisions: steel balls (almost all kept), a car crumple zone (deliberately spent), mud on a wall (all spent). The sorting isn’t random — material stiffness decides the split.

Why is this happening at all? Why is momentum ALWAYS conserved but KE not? Momentum is a vector: internal collision forces cancel in opposite pairs regardless of deformation. Kinetic energy is a scalar with no direction to cancel — it can flow into heat and stay there. Direction gives momentum its indestructibility; energy’s freedom to become heat is exactly what crumple zones exploit to save lives.

▶ Recap card — save for revision week

momentum survives EVERY collision — no exceptions
elastic: KE survives too (ideal bounces)
sticking collision: maximum KE death
equal masses + elastic = velocity swap
light body hitting heavy body bounces back

← Part 5: Power and Efficiency: How FAST You Can Do the Work Part 7: Springs and Vertical Circles: Energy in Two Classic Stages →

Frequently Asked Questions

What should you know about The Survivor: Momentum?

Momentum (p = mv) is conserved in every collision — elastic, inelastic, or perfectly inelastic — because internal forces between the colliding bodies are equal and opposite (Newton’s third law) and act for the same time, so their momentum changes cancel exactly. To apply it: assign a positive direction, write every velocity with its sign, and set total (mass × velocity) before = total after. It works even when energy is lost to heat, sound, and deformation, which is why it is the first equation you write in any collision problem.

What should you know about The Casualty: Kinetic Energy?

KE (½mv²) is a scalar with no direction to cancel, so crash forces can permanently convert it into heat, sound, and deformation. Only in elastic collisions does KE survive intact; in inelastic collisions part of it dies, and in perfectly inelastic collisions the loss is maximum. The exam workflow: find speeds using momentum conservation first, then audit the energy before and after to calculate the loss.

What should you know about Equal Masses in Elastic Hits: The Neat Swap?

When two equal masses collide elastically (one possibly at rest), they simply exchange velocities — the moving mass stops and the other leaves with the original speed. Newton’s cradle demonstrates this repeatedly. Memorise it as a shortcut: it saves you solving two simultaneous equations in the exam. Note it applies only when masses are equal AND the collision is elastic — check both conditions before using it.

What should you know about Solved Examples?

The examples build a repeatable method: (1) write momentum conservation with signs to find final velocities — e.g. (2×6) + 0 = (2+4)v gives v = 2 m/s for the sticking collision; (2) audit energy separately to find what died — 36 J before, 12 J after, so 24 J went to heat and dent; (3) for elastic unequal-mass hits, solve momentum and KE together, then interpret the signs: the 1 kg ball rebounds at −2 m/s while the 3 kg ball moves forward at +2 m/s. Two ledgers, two questions — momentum for speeds, energy for losses.

What should you know about This Physics in Your Daily Life?

Car crumple zones are deliberate KE-killers: bending metal absorbs crash energy over distance so your body doesn’t. Newton’s cradle is the equal-mass elastic swap in metal. Cricket and tennis ‘sweet spot’ hits are the most elastic — the ball leaves fastest because least energy is wasted in vibration. Railway coupling at hump yards is the perfectly-inelastic lab, with wagons sharing one speed after locking. Airbags and helmets extend collision time — same momentum change, gentler force — the impulse idea saving lives.

Springs and Vertical Circles: Energy in Two Classic Stages

In one line: Springs and Vertical Circles — exam-ready notes in one glance.

In one line: JEE/NEET Physics · Work, Energy & Power series · Part 7 of 8 · All parts →✪ Key points — the 30-second versionSprings obey Hooke's law: pull force =.

JEE/NEET Physics · Work, Energy & Power series · Part 7 of 8 · All parts →

✪ Key points — the 30-second version

Springs obey Hooke’s law: pull force = stiffness × stretch (F = kx)
Vertical circles: minimum top speed = √(gR) (gravity supplies the whole inward push)
Energy solves both: ½kx² ↔ KE ↔ mgh trades
Water in a rotating bucket doesn’t fall — the circle’s demand holds it
In fact, a pail, a plane looping, a satellite: one rule, √(gR)

Swing a bucket of water over your head — the water stays in. Loop a plane — passengers are pushed into seats, not belts. Moreover, both are one rule about the minimum speed at the top of a vertical circle. Part 7 of the Work, Energy & Power series : two classic energy stages every exam loves.

In this card.

Hooke’s law, simply.
Therefore, the vertical circle’s top point: the weak link.
The √(gR) rule, derived.
Energy connects the levels.
Solved examples.
Common mistakes.
Meanwhile, this physics in your daily life.
Practice set.
Recap.

Hooke’s Law, Simply.

The vertical circle: the TOP is the weak link — gravity alone must supply the whole inward push, giving the minimum speed √(gR)

TOP: min speed √(gR)
gravity pulls toward centre (helper)

BOTTOM: needs √(5gR)
gravity opposes (must climb out)

release height
h = 2.5R

pull-back force = stiffness × stretch (F = kx). double the stretch, double the pull — and the stored energy squares (½kx², Part 3)

Letter.	What it means (plain words).	Value / unit.
F.	the force the spring pulls back with.	N.
k.	stiffness — newtons per metre of stretch.	N/m.
x.	stretch (or squeeze) from natural length.	m.

Combined with energy: stretch stores ½kx², and releasing converts it to KE. As a result, this pairing (F = kx to find forces. ½kx² to find energy) solves every spring question.

The Vertical Circle’s Top Point: The Weak Link.

In other words, in a vertical circle. Meanwhile, the top is the danger point — gravity pulls you toward the centre (helping the circle) and speed is lowest there (energy spent on climbing). In fact, the question: how slow can you go at the top and still keep the circle?

The √(gR) Rule, Derived.

Notably, at the top, gravity pulls down — straight toward the centre. Meanwhile, in the most desperate case, gravity alone supplies the entire inward push the circle demands :

mg = mv²/R → v_top = √(gR). the absolute minimum top speed — any slower, the circle fails and you drop

Below √(gR), gravity wants more inward pull than the circle’s path can provide — the object leaves the circle (water leaves the bucket). Indeed, at or above it, the track/rotation holds. Meanwhile, one number, universal: bucket, plane, rollercoaster, satellite (whose ‘circle never fails’ because it’s always in free fall).

Energy Connects the Levels.

To find the minimum launch speed at the BOTTOM for a full loop: bottom speed must be enough to climb 2R and still have √(gR) at top. Energy: ½mv_b² = ½m(gR) + mg(2R) → v_b = √(5gR) — the famous √5. Sibling of Part 4’s 2.5R height rule (they’re the same statement, one in speeds, one in heights).

Solved Examples.

✎ Easy — Hooke. A spring (k = 400 N/m) stretched 5 cm. Pull-back force and stored energy?

Force: 400 × 0.05 = 20 N. Energy: ½(400)(0.05²) = 0.5 J.

Note: force linear (20 N), energy quadratic — different books.

Answer: F = 20 N; E = 0.5 J

✎ Exam level — the bucket. Minimum speed at the top of a 1 m vertical circle (g = 10)?

√(gR): √(10 × 1) ≈ 3.16 m/s .

Feel it: one full turn per ~2 seconds — that’s why you swing a bucket briskly, not lazily.

Answer: ≈ 3.16 m/s

✎ JEE level — the √5 launch. A bead on a frictionless vertical loop (R = 0.8 m). Minimum bottom speed to complete the loop (g = 10)?

Energy route: ½v_b² = ½(gR) + 2gR → v_b = √(5gR) = √(5×10×0.8) = √40.

v_b ≈ 6.32 m/s.

Cross-check with Part 4: release height needed = 2.5R = 2 m → v from 2 m drop = √(2×10×2) = √40 ✔ — speeds and heights tell the same story.

Answer: v_b = √(5gR) ≈ 6.32 m/s

⚠ Mistakes students make — and how to avoid them.

Using √(gR) as the BOTTOM speed. It’s the TOP minimum. Bottom needs √(5gR).
Forgetting gravity helps at the top. Specifically, at the circle’s top, gravity points toward the centre — it’s an ally. Indeed, at the bottom, it’s opposition (the track must push extra).
Centimetres in ½kx². 5 cm = 0.05 m, always — the eternal spring trap.
Getting the tension wrong at the top. Similarly, at minimum speed the track/string pushes (or pulls) with ZERO extra force — gravity does it all. That’s the meaning of √(gR).

This Physics in Your Daily Life.

◎ This physics in your daily life.

Overall, the bucket trick works exactly when your hand-side speed beats √(gR) — feel it fail as you slow: water falls from the top.
Rollercoaster loops are engineered above √(5gR) with safety margin — the screams at the top are physics holding you in.
Washing machine spin cycles: the drum spins clothes at speeds where water ‘can’t stay’ in the fabric — it leaves through the holes tangentially. √(gR) logic, laundry edition.
Pilots looping aircraft feel ‘g-force’ at the loop’s BOTTOM (extra push needed) and lightness at the top — the vertical circle’s asymmetry, worn as body weight.
Consequently, every trampoline bounce is F = kx catching you and ½kx² returning you — this card’s two halves in one mattress.

Practice set (answers hidden — try first).

(NEET-level) Spring k = 200 N/m, x = 10 cm. Force:.

F = 200 × 0.10 = 20 N.

(JEE Main-level) Minimum top speed in a 0.4 m vertical circle (g = 10):.

√(10 × 0.4) = 2 m/s.

(JEE Main-level) Minimum bottom speed for the same loop:.

√(5 × 10 × 0.4) = √20 ≈ 4.47 m/s.

(Concept) At the top at minimum speed, the string’s tension is:.

Zero — gravity supplies the entire inward push.

(NEET-level) Doubling a spring’s stretch multiplies its stored energy by:.

4 (x²).

🧠 Memory tricks & everyday anchors — the 20-second revision

🧠 Chant: ‘top is √gR, bottom is √5gR, height is 2-and-a-half R’.
🧠 Gravity flips roles: helper at the top, opponent at the bottom of a vertical circle.
🏠 Daily: swing a bucket briskly — you’re personally verifying √(gR).
🏠 Daily: the washing machine’s spin cycle is water failing to keep the circle.
🔁 F = kx (linear) vs ½kx² (squared)
🔁 v_top(min) = √(gR): gravity alone supplies the push
🔁 v_bottom(min) = √(5gR)

One idea, three doors — open whichever clicks for you

Same concept (why spring energy is ½kx² and loops trade height for speed), three different ways of seeing it. If one door confuses you, try the next — at least one will stick.

Door 1 · The story way

A trampoline catch: the deeper you sink it, the HARDER it pushes back — resistance grows with stretch. That growing resistance is why spring energy isn’t just force × distance: the average force over the stretch is only half the final force.

Door 2 · The numbers way

Stretch a spring (k = 100 N/m) by 0.2 m: final force 20 N, average force 10 N, energy = 10 × 0.2 = 2 J = ½kx². The ½ is the triangle’s area — force ramping from 0 to 20 over the stretch. Vertical loop: at the top, mg = mv²/r; solve and v_top = √(gr) — the minimum loop speed.

Door 3 · The picture way

Draw force versus stretch: a straight diagonal line from zero. The stored energy is the shaded triangle under it — base x, height kx, area ½kx². For the loop: a circle with speed arrows — long at the bottom, shortest at the top — and a height ledger trading between them.

Why is this happening at all? Why the ½? Because spring force ramps linearly from zero, so the average over the pull is half the peak — energy is the area under the force curve, and a ramp to a peak only fills half the rectangle. Why do loops work? Because gravity is conservative: every metre of height lost is refunded exactly as speed, so only the difference matters.

▶ Recap card — save for revision week.

Hooke: F = kx (force linear); energy: ½kx² (squared)
top of vertical circle: minimum v = √(gR)
bottom launch for a full loop: √(5gR)
height equivalent: 2.5R (same rule, in heights)
at minimum top speed, gravity supplies the whole inward push

← Part 6: Collisions: The Great Sorting — What Survives, What Dies Part 8: The Finale: Energy in the Real World, and the Complete Formula Card →

Frequently Asked Questions

What should you know about Hooke's Law, Simply?

What should you know about The Vertical Circle's Top Point: The Weak Link?

What should you know about The √(gR) Rule, Derived?

Furthermore, at the top, gravity pulls down — straight toward the centre. Meanwhile, in the most desperate case, gravity alone supplies the entire inward push the circle demands : Below √(gR). Gravity wants more inward pull than the circle’s path can provide — the object leaves the circle (water leaves the bucket). In other words, at or above it, the track/rotation holds. Meanwhile, one number, universal: bucket, plane, rollercoaster, satellite (whose ‘circle never fails’ because it’s always in free fall).

What should you know about Energy Connects the Levels?

What should you know about Solved Examples?

Note: force linear (20 N), energy quadratic — different books. ✔ Using √(gR) as the BOTTOM speed. It’s the TOP minimum. Bottom needs √(5gR). Forgetting gravity helps at the top. Likewise, at the circle’s top, gravity points toward the centre — it’s an ally. Meanwhile, at the bottom, it’s opposition (the track must push extra).

Examiner’s Corner: The Three Traps

Trap 1: Forgetting the pivot is a point, not a support

In vertical-circle problems the top of the circle is the stress point: the minimum speed there is sqrt(gR) for a string or a track contact, because gravity alone must supply the centripetal force. Students who draw the free-body diagram at the bottom and never check the top lose the whole question. At the bottom, the tension or normal force is maximum: T = mv²/r + mg; at the top, minimum: T = mv²/r − mg. The difference between the two readings is 6mg for a full swing at critical speed — a favourite numerical.

Trap 2: Mixing spring energy with gravitational potential

A spring released from natural length while a mass falls does not convert all gravitational potential into spring energy unless the question says equilibrium. At maximum stretch the mass is momentarily at rest but acceleration is not zero — it is (kx minus mg)/m. Equilibrium stretch, where net force vanishes, is mg/k, exactly half the maximum stretch when dropped from rest. That factor-of-two distinction generates endless single-digit questions.

Trap 3: SHM disguised as circular motion

Uniform circular motion projected on any diameter is SHM. So a spring-block oscillation problem can be solved by energy conservation or by the SHM toolkit; both must agree. When an exam gives a spring constant and a mass and asks for the time period, T = 2*pi*sqrt(m/k) is the one-line route — do not re-derive from force equations under time pressure.

Worked Mini-Set

Sources and further reading

Energy in the Real World: The Formula Card

In one line: JEE/NEET Physics · Work, Energy & Power series · Part 8 of 8 · All parts →✪ Key points — the 30-second versionVariable forces: work = area under the.

In fact, JEE/NEET Physics · Work, Energy & Power series · Part 8 of 8 · All parts →

✪ Key points — the 30-second version

Moreover, variable forces: work = area under the force-distance graph
Therefore, energy curves: valleys = stability, hills = instability
Meanwhile, rockets, humans, engines: everyone obeys the same energy ledger
As a result, efficiency chains explain the entire energy economy
In other words, complete chapter formula card at the end

Notably, a rocket burns tonnes of fuel, your body runs a marathon on a plate of rice. Meanwhile, a dam lights a city — three systems, one ledger. In fact, the finale of the Work, Energy & Power series handles the advanced leftovers and hands you the complete formula card.

In this card

Variable forces: the graph trick
Energy curves: reading stability
The human engine
The energy economy
Solved examples
Common mistakes
Indeed, this physics in your daily life
Practice set
Recap + formula card

Variable Forces: The Graph Trick

Specifically, w = Fd assumed constant force. When the force changes (springs! Moreover, air drag!), plot force vs distance — the work is the area under the graph. Moreover, spring work ½kx² is exactly the triangle under F = kx: ½ × base × height = ½ × x × kx. One picture unifies every variable-force case.

Energy Curves: Reading Stability

Similarly, plot a body’s PE against position. Meanwhile, valleys = stable equilibrium (pushed away, it rolls back — a ball in a bowl). Therefore, hills = unstable (a pencil on its tip — any nudge and it leaves). Flat = neutral (a ball on a table). And a small wiggle at a valley’s bottom is automatically simple harmonic motion — the bridge into the next series, Oscillations.

The Human Engine

Overall, your body runs at ~100 W idle, ~400 W walking, ~1,000 W sprinting (elite cyclists touch 1,500 W bursts). Indeed, a day’s food ~9 MJ — roughly a 100 W bulb burning 24 hours. You are, quite literally, a moderately powerful heat engine with excellent snack logistics.

The Energy Economy

Consequently, chemical (fuel/food) → heat → motion/electricity, with losses at every step. Meanwhile, a power plant: fuel → steam → turbine → electricity ≈ 40% max. An EV battery-to-wheel ≈ 85%; incandescent bulb: 5% light, 95% heat. Every ‘energy crisis’ discussion and star-rating sticker is this chapter at civic scale.

Solved Examples

✎ Easy — graph work. A force grows linearly from 0 to 50 N over 4 m. Work?

Furthermore, area under the line = triangle = ½ × 4 × 50 = 100 J .

Likewise, cross-check: average force 25 N × 4 m = 100 J ✔

Answer: 100 J

✎ Exam level — curve reading. A PE curve has a valley at x = 2 m. At the valley floor, the force on the body is:

In short, force = the curve’s slope — at a valley’s floor, slope = 0 → zero force (equilibrium). Meanwhile, displaced either way, the slope pushes it back — that’s stability.

Answer: zero force; stable — it returns

✎ JEE level — full chain. A 60% efficient motor pumps 5,000 kg of water up 12 m each minute (g = 10). Electric power drawn?

Subsequently, useful: mgh/t = 5,000×10×12 ÷ 60 = 10,000 W.

Drawn: 10,000 ÷ 0.6 ≈ 16.7 kW.

In fact, the 6.7 kW gap = motor heat — efficiency is always a heat story.

Answer: ≈ 16.7 kW

⚠ Mistakes students make — and how to avoid them

Work = area under force-distance graph — count squares or use triangle/rectangle shapes. Meanwhile, never assume constant force when told it varies.
Valley vs hill confusion. Indeed, valley = stable (returns), hill = unstable (leaves). Draw the ball; feel the answer.
Efficiency multiplied wrong direction. Specifically, input = useful ÷ efficiency (bigger); output = input × efficiency (smaller). Check with the ‘must be < 100%' rule.
Human power overestimated. Similarly, a human sustains ~100-150 W, peaks ~1,000+ W. We’re light bulbs, not engines.

This Physics in Your Daily Life

◎ This physics in your daily life

Overall, fuel prices, star ratings, EV debates, climate targets — all public arguments about efficiency chains. This chapter is the literacy behind the headlines.
Consequently, your breakfast is a power contract: ~2. 000 food-calories ≈ 8.4 MJ ≈ a 100 W machine’s daily supply — you budget energy like any engine.
Furthermore, mountain roads zigzag because engines (fixed power) trade distance for force on climbs — switchbacks are P = Fv carved into geography.
Bungee cords and climbing ropes are engineered force-distance curves: they stretch to extend stopping distance. Softening the force peak — the area under the graph, saving spines.
Grid-scale batteries and pumped lakes buy energy cheap, store it (PE!), sell it dear — the ledger, monetised at national scale.

What	Formula	Remember
Work	W = Fd·cosθ	perpendicular = zero; against motion = negative
Kinetic energy	½mv²	square! double speed ×4
Work-energy theorem	W_total = ΔKE	before/after only — path-free
Height PE	mgh	choose one zero level
Spring PE	½kx²	stretch squared; metres!
Energy conservation	KE + PE = constant (gravity/springs)	friction leak = F·d → heat
Drop speed	v = √(2gh)	no mass anywhere
Loop minimums	v_top = √(gR); v_bottom = √(5gR); h = 2.5R	gravity helps at the top
Power	P = W/t = Fv	watts; 1 hp = 746 W
Efficiency	useful ÷ input	always < 100%
Collisions	momentum always survives	sticking = max KE loss; equal-mass elastic = swap
Variable force	work = area under F-d graph	½kx² is the triangle
PE curves	valley = stable, hill = unstable	slope = force

Practice set (answers hidden — try first)

(NEET-level) Force rises linearly 0→30 N over 6 m. Work:

Triangle: ½ × 6 × 30 = 90 J.

(Concept) A PE curve’s hill-top is what kind of equilibrium:

Unstable — any nudge and the body leaves.

(JEE Main-level) A 75% motor delivers 3 kW useful. Input power:

3 ÷ 0.75 = 4 kW.

(Concept) Why do switchback mountain roads exist?

Fixed engine power: trading distance for climbing force (P = Fv) — geography applying this chapter.

(JEE Main-level) A ball dropped from h on a spring (k): maximum compression x satisfies:

mgh = ½kx² → x = √(2mgh/k).

🧠 Memory tricks & everyday anchors — the 20-second revision

🧠 Graph chant: ‘the area under the force curve IS the work’.
🧠 Three roots to remember: √(2gh) drop, √(gR) loop-top, √(5gR) loop-bottom.
🏠 Daily: you are a ~100 W appliance that runs on rice — the ledger applies to bodies too.
🏠 Daily: every star rating and fuel-price headline is this chapter at civic scale.
🔁 work = area under F-d graph
🔁 valley/hill on PE curve = stable/unstable
🔁 √(2gh), √(gR), √(5gR) — the three famous roots

One idea, three doors — open whichever clicks for you

Same concept (why energy thinking beats force thinking), three different ways of seeing it. If one door confuses you, try the next — at least one will stick.

Door 1 · The story way

Force thinking asks ‘what pushes what, right now?’ — and gets tangled in vectors. Energy thinking asks ‘what do I have at the start, what at the end?’ — and skips the middle entirely. For any question about speeds, heights, and stopping distances, energy accounting is the shortcut that never lies.

Door 2 · The numbers way

Braking distance: friction force F stops a car of mass m at speed v. Energy view: ½mv² = F×d → d = mv²/2F. Double the speed: distance QUADRUPLES. That one line explains every highway-safety poster ever printed.

Door 3 · The picture way

Picture a ledger with two columns — START (KE + PE + work in) and END (KE + PE + work out). Draw a line under both: they must match. However complicated the middle (bumps, brakes, bends), the ledger doesn’t care about the journey, only the totals.

Why is this happening at all? Why is it allowed to skip the middle? Because for conservative forces the intermediate path contributes nothing to the totals — work depends only on endpoints. As long as nothing non-conservative intervenes, the endpoints ARE the whole story; when friction intervenes, it enters the ledger as a single number. Either way: totals in, totals out.

▶ Recap card — save for revision week

variable force: work = area under the force-distance graph
PE curve: valley stable, hill unstable; slope = force
v = √(2gh), √(gR), √(5gR) — the chapter’s famous roots
efficiency chains: input = useful ÷ efficiency
momentum survives collisions; energy often dies

← Part 7: Springs and Vertical Circles: Energy in Two Classic Stages

Frequently Asked Questions

What should you know about Variable Forces: The Graph Trick?

W = Fd assumed constant force. When the force changes (springs! air drag!), plot force vs distance — the work is the area under the graph. Spring work ½kx² is exactly the triangle under F = kx: ½ × base × height = ½ × x × kx. One picture unifies every variable-force case.

What should you know about Energy Curves: Reading Stability?

Plot a body’s PE against position. Valleys = stable equilibrium (pushed away, it rolls back — a ball in a bowl). Hills = unstable (a pencil on its tip — any nudge and it leaves). Flat = neutral (a ball on a table). And a small wiggle at a valley’s bottom is automatically simple harmonic motion — the bridge into the next series, Oscillations.

What should you know about The Human Engine?

Your body runs at ~100 W idle, ~400 W walking, ~1,000 W sprinting (elite cyclists touch 1,500 W bursts). A day’s food ~9 MJ — roughly a 100 W bulb burning 24 hours. You are, quite literally, a moderately powerful heat engine with excellent snack logistics.

What should you know about The Energy Economy?

Chemical (fuel/food) → heat → motion/electricity, with losses at every step. A power plant: fuel → steam → turbine → electricity ≈ 40% max; an EV battery-to-wheel ≈ 85%; incandescent bulb: 5% light, 95% heat. Every ‘energy crisis’ discussion and star-rating sticker is this chapter at civic scale.

What should you know about Solved Examples?

Area under the line = triangle = ½ × 4 × 50 = 100 J . Cross-check: average force 25 N × 4 m = 100 J ✔ Fearing graphs. Work = area under force-distance graph — count squares or use triangle/rectangle shapes. Never assume constant force when told it varies. Valley vs hill confusion. Valley = stable (returns), hill = unstable (leaves). Draw the ball; feel the answer.

Work Done: When a Force Actually Achieves Something

Work Done in Physics: Definition, Formula, and Examples

The Simple Idea: Force × Movement

What Each Letter Means

The Angle Rule: Only the Matching Part Counts

The Zero-Work Surprises

Positive and Negative Work

Solved Examples

This Physics in Your Daily Life

Practice set (answers hidden — try first)

Frequently Asked Questions

What should you know about The Simple Idea: Force × Movement?

What should you know about The Angle Rule: Only the Matching Part Counts?

What should you know about The Zero-Work Surprises?

What should you know about Positive and Negative Work?

What should you know about Solved Examples?

References & authoritative sources

Kinetic Energy and the Work-Energy Theorem

Kinetic Energy and the Work-Energy Theorem Explained

Moving Energy, Simply

The Work-Energy Theorem: The Great Shortcut

The v² Law and Road Safety

Solved Examples

This Physics in Your Daily Life

Practice set (answers hidden — try first)

Frequently Asked Questions

What should you know about Moving Energy, Simply?

What should you know about The Work-Energy Theorem: The Great Shortcut?

What should you know about The v² Law and Road Safety?

What should you know about Solved Examples?

What should you know about This Physics in Your Daily Life?

References & authoritative sources

Potential Energy: Stored Work, Ready to Strike

Potential Energy Explained: Stored Work Ready to Strike

Height Energy: mgh

Spring Energy: ½kx²

Conservative: Path Never Matters

The PE ↔ KE Trade

Solved Examples

This Physics in Your Daily Life

Practice set (answers hidden — try first)

Frequently Asked Questions

What should you know about Height Energy: mgh?

What should you know about Spring Energy: ½kx²?

What should you know about Conservative: Path Never Matters?

What should you know about The PE ↔ KE Trade?

What should you know about Solved Examples?

References & authoritative sources

Conservation of Energy: The Universe's Perfect Bookkeeping

Conservation of Energy: How the Universe Keeps Perfect Books

The One Rule

The Famous Result: v = √(2gh), No Mass Anywhere

When Friction Leaks the Ledger

The Pendulum’s Endless Trade

Solved Examples

This Physics in Your Daily Life

Practice set (answers hidden — try first)

Frequently Asked Questions

What should you know about The One Rule?

What should you know about The Famous Result: v = √(2gh), No Mass Anywhere?

What should you know about When Friction Leaks the Ledger?

What should you know about The Pendulum's Endless Trade?

What should you know about Solved Examples?

References & authoritative sources

Power and Efficiency: How FAST You Can Do the Work

Power and Efficiency: How Fast You Can Get Work Done

Power, Simply

The Two Workhorse Formulas

Efficiency: Nothing Is 100%

Solved Examples

This Physics in Your Daily Life

Practice Set (answers hidden — try first)

Frequently Asked Questions

What is power in physics, in simple words?

What are the typical power ratings of everyday devices?

Why do cars slow down when climbing a hill?

How efficient are common machines?

What is the most common power-and-efficiency exam mistake?

How does the electricity bill relate to power and energy?